STUDENT AND TEACHER AFFECT

The goals of enjoying mathematics and being persistent are of
paramount importance to a students success and should be
primary considerations in the design of class activities. The goals
of a research experience are for students to engage in lengthy,
complex projects; to take risks and grapple with deep ideas; and to
be sufficiently satisfied with the experiences so that they want to
face such difficulties again in their lives. Realizing these goals
requires tenaciousness and a tolerance for frustration. Students
(and adults) can become paralyzed, depressed, or antagonistic when
faced with open-ended, incompletely defined tasks. While lessons
should gradually increase in the level and length of challenges
involved, no amount of easing in can eliminate the symptoms just
cited. It is essential to make clear to the students what they will
face, how they may feel, why you are asking them to experience
certain frustration, and how you will support them throughout the
process.

Student Expectations

Many students believe that a teachers job is to explain
everything for themto turn complicated ideas into simple,
predigested ones. Students and teachers alike are uncomfortable
with the possibility that a teacher could answer a question with "I
dont know." When students are engaging in real
researchwhen they are immersing themselves in the
doing of a disciplineyour role is not to be the
Grand Explainer. It is important to tell your students that your
job is to help them find interesting avenues for investigation and
to help them learn how to overcome the many intellectual obstacles
they will encounter in their work. Your job is to provide a
framework for approaching new situations, to ask the questions the
students need to learn how to ask themselves. A Making Mathematics
mentor encouraged their teacher mentee with the following note:

So much of the stuff you write about"they no longer needed
me to validate their work... they saw how much more complicated the
problem became..."thats all so important in understanding
how mathematics really works. If youre trying to work on a problem
thats new, no one can validate your work because
youre the first one to try it! You have to
figure it out, convince yourself, and then convince others. I also
like that you reinforced how much progress they had made since starting
to work on the project; kids dont often see that kind of progress
around a single problema noticeable, measurable increase in
what theyre able to do with it. Thats really cool.

Why should students want to undertake such a perilous voyage
with guarantees of hardship? Because it is in the overcoming of
those impediments that deep understanding emerges. To demonstrate,
ask students to add 3 + 5 and then ask them how much they learned
from that problem. Learning occurs when we try new challenges that
force us to discover new ideas or make new connections between old
ideas.

Many students think that a teacher who will not explain
everything is leaving them to their own devices. On a regular
basis, remind your students (or a recalcitrant individual) that you
do not expect them to sink or swim on their own. Encourage (or
shanghai) them to come to ask questions about a problem. Some
students have difficulty assessing when they are really stuck, some
think they are always stuck (and ask for help too quickly), and
others think asking for help is wrong. It is sometimes difficult to
distinguish for the students and oneself where the line between
increasing independence begins and avoiding floundering ends. Err
on the side of giving extra help, so long as that help models the
internal dialogue that problem-solvers need. When a student is
stuck, share with them the questions that you would ask yourself in
order to solve the problem. Ask your students to think about how
your questions are a response to a particular situation and
encourage them to ask themselves questions about the problems that
they tackle (see Getting Stuck, Getting
Unstuck).

Of course, none of us is going to discover every piece of
mathematics that we will need to solve the problems that we will
encounter. If a student is stuck and the only, or best, path to
progress seems to involve a topic that they could learn, it makes
sense to teach them those new ideas and techniques or to lend them
a book so that they can teach themselves.

Advice For Long-Term Class Research

Students are much more open to meeting unsettling expectations
if they, in turn, have some control over how the course is
structured. Students are understandably fearful of failure.
Negotiating due dates for major assignments can help students feel
confident that they will have time to do the assignment well. It is
also reasonable to offer extensions when students can demonstrate
that they worked hard on a problem but would like to try to make a
little more progress. Requiring that extensions be requested at
least a day in advance discourages last-minute efforts and adds the
minimal requirement that students realize when they are not
progressing as quickly as they had hopeda step students need
to take when monitoring the progress of lengthy projects. Evidence
of effort should consist of more than wracked brainsthey
should provide substantial logbook work.

The above policies reduce the stress associated with research
deadlines when these deadlines may not conform to the particular
time needs of a student and his or her project. Creative
breakthroughs are difficult to produce on a schedule, but, with
persistence, they should occur, and it is important to reduce the
sense of risk associated with these uncertainties. Weekly
touch-base sessions that allow students or research groups the
chance to provide updates on their progress help students avoid
last-minute "cramming". Additionally, these sessions give the
teacher a chance to ask clarifying questions about the work
undertaken thus far and reduce the stress some students feel when
asking for individual help outside of class. Written updates, such
as rough drafts or short progress reports from the students, are
also useful for scaffolding students use of time.

Student Interests

Another way teachers give students control of a course is
through the attention we pay to their ideas, questions, and
interests. This attention is central to meeting the goals that
students be investigative, make connections, and identify topics
for exploration. Students only embrace these goals if the
sought-after behaviors get the response they deserve, namely, class
discussions and assignments that move in the direction of their
thoughts and interests. For example, in one of our classes, as a
result of student questions, the students spent two weeks studying
game theory and the Prisoners Dilemma. This seeming
digression spurred many students on to ask for readings on
mathematics topics of individual interest to them.

Although the thought of having students choose course content
may disturb some, this choosing is a clear indication that students
are developing particular mathematics interests and actually want
to do math. However, this approach means that heavily detailed
syllabi become either useless (as you would need to update the
syllabus daily, or ignore students comments that did not head
in the pre-planned direction) or destructive (students who see
content-laden syllabi quickly shut down their own initiative and
mathematical problem-posing). Just as teachers have their own
mathematical aesthetics, so do students. We all have favorite
topics and activities that get us really excited. Both variety and
flexibility are essential to maintaining this built-in source of
motivation.

Change Takes Time

The development of persistence in both individuals and a class
requires careful planning. It is important that students have some
early successes, but these need not always be immediate or
complete. Challenges of greater length and difficulty should be
introduced gradually, and students need to have most of the skills
and understandings required to solve a problem (with just one or
two gaps, which may be the sought-after goal of the activity). Work
on projects follows rhythms (of effort, insight, and types of
tasks) that are not present in shorter exercises and to which it
takes time to adapt. Both the teacher and students need to be
patient with these changes.

If you are only spending a few class periods introducing
research ideas (e.g., using the Introductory
Explorations activity, a warm-up problem from a project, or Trains), you may
discover the frustration that students have when they are stymied
by a problem and you may not have the time to help them master all
of the emotional and mathematical understandings that will make
them want to persist with research. Try to pick content that is
familiar to students so that the focus can be on the research ideas
themselves (e.g., the role of conjectures). Have reasonable
expectations for a first research experience. Return to research
activities periodically so that students will have practice that
develops their understanding of the research process and builds
their confidence that they can stick with a problem until they have
made progress with it.

Teacher Affect

The most important determinant of the students affect will
be the teachers. We must be excited about mathematics and
share that enthusiasm unashamedly with our students. Students will
jump in when they see that you are interested in their questions,
work on their problems, and bring them articles and books on topics
of interest. You will be most effective if you are willing to get
happily perplexed by difficult problems in front of your classes.
No one expects figure skaters to divine the proper position for a
leap; they are shown one in action. Similarly, teachers as coaches
must be mathematicians in front of their students, willing to take
an occasional "fall on the ice." Students will be impressed if you
can show your excitement about the process of exploring an
intriguing conjecture whose truth you do not know. If you get
stuck, continue to work on the problem on your own and bring in any
results or your latest dead-ends until you have made some progress.
You can also consult a colleague or book and cite your research. Be
sure to allow students to tackle problems even if you do not know
the answer in advance. If kids see that you are willing to let go
of a safety net, then they will be too.

The best way to prepare for doing mathematics investigations
with students is to do mathematics research oneself. Doing research
hones our ability to identify different choices and make decisions
throughout the research process. It also makes us more sensitive to
the emotional tumult that open-ended research evokes in students
and adult mathematicians alike. Consider the following comments
from two Making Mathematics teachers working with other teachers on
a research question via email. The first quote is a wonderfully
honest description of a fairly typical reaction to a new
problem:

After receiving the first communication I have to tell you...I was
ready to quit. I read the assignment and my eyes crossed. Panic. Overwhelmed.
Then I thought of all my students (8th grade, critical year for self-concept
as a math student) who have to live this all the timedont
understand as quickly as their peers, cant get it in the first
readingand I decided to continue. If nothing else (and, of course,
there will be a lot more), my empathy meter will be higher next year.
So, I read and reread the email, checked the suggested links, started
to get a handle on it and the panic subsided. Im here for the
duration.

This teacher went on to pose many interesting problems that
expanded the groups research agenda. Emotions while doing
research often continue to swing wildly even after that first
fearful start. Consider this teachers roller coaster:

Early on: Im getting a real feel of what it is like to wrestle
with a chart of numbers and try to make some sense of it. My wife
keeps yelling to me from downstairs "Its summer; put the math
away!" I dont know how much longer I can keep my enthusiasm
up without a breakthrough. For now, Im going to give it a rest.
I have been noticing how I go about dealing with things though. I
really like to make use of technology where I can (realizing it doesnt
constitute a proof; but its sure nice to have lots of data to
work with.) Im also pretty visual. I started using colored pencils
right away.

Two days later after a breakthrough: Anyhow, I felt I accomplished
something today. I too feel guilty when Im not inspired to move
on or when I get discouraged. I keep thinking, how will I get my students
to stay on task if I wander myself?

A few days later: I started working with the actual numbers
to see if I could find a functional relationship (I tend to head towards
functions right away, because I do a lot with them in the courses
I teach.) This just overwhelmed me. I made two columns "Row #" and
"Number of entries that are divisible by 3." I could see this wasnt
going to be easy for me to do. In short, Im stuck and feeling
lost. Im OK with experiencing these emotions, because I want
to be able to empathize with my students. But I must say, Im
feeling like a wimp right now, because I want to say this is too hard.

This teacher made a number of discoveries that he was able to
prove. Typical of successful research, his progress often followed
periods of intense immersion coupled with lulls that allowed his
thoughts to settle.

Doing research does not always make us feel more confident about
our mathematical skills. The more challenges we take on, the more
we learn about the extraordinary depth and breadth of the
subjectwe learn how much we do not know and how difficult the
field can be. Strive to let that knowledge inspire you with the
possibilities that lie ahead for you, your students, and for
mathematics itself. Teachers often fear being asked questions in
class that they cannot answer, but being experienced in mathematics
and being all-knowing are not the same thing.

I find it liberating to be able to answer a question with "I
dont know!" I am not pretending not to know, so I am that
much more able to model good problem solving. I think the lack of
certainty in such circumstances is very excitingit is real
math! Science experiments where the teacher always knows what
should happen can seem forced. Real science is about new questions,
failed experiments, and messy data. Mathematics should be full of
the same dead ends and surprising discoveries. The alternative to
admitting our occasional ignorance is saying that mathematics is a
small set of "school" skills that we have mastered and that our
students must learn. That is not a fun, intellectually interesting,
or educationally useful choice.

Final Thoughts

There is an additional benefit and challenge posed by the strong
feelings of joy and frustration elicited by open-ended research.
They force us to acknowledge more of each students
personality. We have to accept their individual responses to
frustration and help them both adjust and learn to cope with these
emotions. This necessity highlights how curricular reform often
forces multiple changes because of the complexity of good teaching
and learning.

One barrier to engaging students in mathematics is the notion,
common in the United States, that our discipline is somehow special
and that real work in mathematics requires a talent that few
possess (see Coping
With Math Anxiety for further discussion). You can help
dissolve this sense of a "high priesthood" by referring to your
students as mathematicians and by helping your class see itself as
a mathematical community in search of discoveries. This language
will complement peer review activities (see Feedback in the Assessment section, and Writing
Conjectures in the Conjectures section) and help
the students recognize that there is an audience for their efforts
beyond their teacher.